A233090 Decimal expansion of Sum_{n>=1} (-1)^(n-1)*H(n)/n^2, where H(n) is the n-th harmonic number.

7, 5, 1, 2, 8, 5, 5, 6, 4, 4, 7, 4, 7, 4, 6, 4, 2, 8, 3, 7, 4, 8, 3, 6, 3, 5, 0, 9, 4, 4, 6, 5, 6, 2, 4, 4, 2, 2, 8, 1, 1, 6, 4, 3, 2, 7, 1, 2, 8, 1, 1, 8, 0, 1, 1, 2, 0, 1, 6, 9, 7, 2, 2, 0, 8, 8, 6, 4, 8, 8, 7, 8, 6, 1, 6, 4, 4, 5, 6, 8, 1, 3, 6, 6, 5, 3, 4, 9, 2, 1, 0, 0, 5, 8, 3, 4, 5, 3, 6, 3

Offset

0,1

References

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 43.

Links

Table of n, a(n) for n=0..99.

R. Barbieri, J. A. Mignaco, and E. Remiddi, Electron form factors up to fourth order. I., Il Nuovo Cim. 11A (4) (1972) 824-864, table II (13)

Philippe Flajolet and Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998), page 32.

Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (C5.15)

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 561.

Formula

Equals 5*zeta(3)/8.

Equals -Integral_{x=0..1} (log(1+x)*log(1-x)/x)*dx. - Amiram Eldar, May 06 2023

Equals Sum_{m>=1} Sum_{n>=1} (-1)^(m-1)/(m*n*(m + n)) (see Finch). - Stefano Spezia, Nov 02 2024

Example

0.7512855644747464283748363509446562442281164327128118011201697220886...

Mathematica

RealDigits[ 5*Zeta[3]/8, 10, 100] // First

Crossrefs

Cf. A002117 (zeta(3)), A197070 (3*zeta(3)/4), A233091 (7*zeta(3)/8), A076788 (alternating sum with denominator n), A152648 (non-alternating sum with denominator n^2), A152649 (non-alternating sum with denominator n^3), A233033 (alternating sum with denominator n^3).

Sequence in context: A093205 A156536 A110191 * A254177 A377609 A021575

Adjacent sequences: A233087 A233088 A233089 * A233091 A233092 A233093

Keyword

nonn,cons

Author

Jean-François Alcover, Dec 04 2013, after the comment by Peter Bala about A233033.

Status

approved